Poles and Zeros for Transfer Functions

This chapter would be complementary to chap. 1 . Zeros are well formulated for room-acoustic transfer functions. Four cases are possible in occurrences of zeros between a pair of adjacent poles. No zeros, a single zero, double zeros on the pole line, and symmetric pair of off-line zeros with respect to the pole line. The cases depend on signs of residues for the adjacent pair of poles. Feedback systems including sound fields in a closed loop would be an example of room-acoustic transfer functions. The condition formulated by poles and zeros of transfer functions would be helpful to understand why the stability of feedback systems may be sensitive to the locations of microphone and loudspeakers in the sound field. Instability may be a consequence of a non-minimum phase zero in the closed loop. Thus, negative feedback systems could be re-formulated by poles, zeros, and inverse filtering. Feedback may be a possible approach also to increasing frequency selectivity or getting to prominent resonance such as designing musical instruments as well as a public address system. The transfer function could be decomposed into all-pass and minimum-phase components according to complex inversion pairs of poles and zeros. Decomposition of a transfer function into the all-pass and minimum-phase systems could be formulated according to the complex inversion pairs of poles and zeros, too.